Two adjacent pipe diameters at the optimal solution in the water distribution network models

In the study of optimization models for the design of water distribution networks, most of the literature has either indicated or explicitly claimed that at the optimal solution each link will consist of at most two pipe segments with adjacent diameters. This paper presents the necessary and sufficient conditions that a given set of pipe diameters is in an optimal solution and as a special case shows that the adjacency property holds if and only if pipe costs are a strictly convex function of a power of pipe diameters.